Ch 11: Impulse and Momentum

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 11: Impulse and Momentum

Problem 21

Chapter 11, Problem 21

Fred (mass 60 kg) is running with the football at a speed of 6.0 m/s when he is met head-on by Brutus (mass 120 kg), who is moving at 4.0 m/s. Brutus grabs Fred in a tight grip, and they fall to the ground. Which way do they slide, and how far? The coefficient of kinetic friction between football uniforms and Astroturf is 0.30.

Verified step by step guidance

Step 1: Identify the type of collision. This is an inelastic collision because Fred and Brutus stick together after the collision. Use the principle of conservation of momentum to determine their combined velocity immediately after the collision.

Step 2: Write the equation for conservation of momentum:

m F_{r} v F_{r} + m B_{r} v B_{r} = (m F_{r} + m B_{r}) v

_f, where

m F_{r}

and

m B_{r}

are the masses of Fred and Brutus,

v F_{r}

and

v B_{r}

are their velocities before the collision, and

v

_f is their final velocity together.

Step 3: Solve for the final velocity

v

_f. Substitute the given values:

m F_{r} = 60

kg,

v F_{r} = 6.0

m/s,

m B_{r} = 120

kg, and

v B_{r} = -4.0

m/s (negative because Brutus is moving in the opposite direction).

Step 4: Determine the direction of motion after the collision. The sign of

v

_f will indicate the direction of their combined motion. If

v

_f is positive, they slide in Fred's original direction; if negative, they slide in Brutus's original direction.

Step 5: Calculate the distance they slide. Use the work-energy principle. The kinetic energy of the combined mass is dissipated by the force of kinetic friction. Write the equation:

\frac{1}{2} (m F_{r} + m B_{r}) v

_f² = f_kd, where

f

_k = μmg is the force of kinetic friction,

μ

is the coefficient of kinetic friction, and

d

is the distance. Solve for

d

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Conservation of Momentum

The principle of conservation of momentum states that in a closed system, the total momentum before an interaction is equal to the total momentum after the interaction. In this scenario, Fred and Brutus collide and stick together, so we can calculate their combined momentum to determine their initial velocity after the collision.

Kinetic Friction

Kinetic friction is the force that opposes the motion of two surfaces sliding against each other. It is calculated using the coefficient of kinetic friction and the normal force. In this case, the coefficient of kinetic friction between the football uniforms and the Astroturf will help determine how quickly Fred and Brutus come to a stop after sliding.

Net Force and Acceleration

Net force is the total force acting on an object, which determines its acceleration according to Newton's second law (F = ma). After the collision, the net force acting on Fred and Brutus will be the kinetic friction force, which will decelerate them as they slide. Understanding this relationship is crucial for calculating how far they will slide before coming to a stop.

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